An NL Fragment for Inclusion Logic Dietmar Berwanger joint work - - PowerPoint PPT Presentation

An NL Fragment for Inclusion Logic

Dietmar Berwanger joint work with Erich Grädel Dagstuhl, June 

Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 1 / 10

Non-reachability of Q NL ∃z(x ⊆ z ∧ ¬Qz ∧ ∀y(Ezy → y ⊆ z)) Winning safety game within Q P-complete Qx ∧ ∃z(z ⊆ x ∧ (V → ∃y(Exy ∧ y ⊆ z)) ∧ (V → ∀y(Exy → y ⊆ z))

Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 2 / 10

Solitaire games and GFP

Solitaire games — only one player has nontrivial moves, in every SCC Inspire restriction to Solitaire GFP:

• negation: only fixed-point-free formulae
• conjunction: one member fixed-point-free

Captures NL on finite structures & more expressive than FO + TC on arbitrary structures Goal: Find the Solitaire-GFP fragment of Inclusion logic Obstacles: closed formulae in team semantics? dual of safety conditions

Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 3 / 10

Flattening

Definition Formula φ ∈ Inc is flat: A ⊧X φ iff A ⊧{s} φ, for all s ∈ X. Lemma: flattening φ(¯ x) is flat iff for any structure A, team X A ⊧X φ(¯ x) ⇐ ⇒ A ⊧X ∃¯ z(¯ x ⊆ ¯ z ∧ φ(¯ z)) ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ φ♭(¯ x)

Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 4 / 10

Te fragment. Take 

F ∶∶= literals, ⊆ atoms φ ∨ η ∃¯ xφ for φ, η ∈ F φ♭ ∧ η ∀¯ xφ♭ with φ♭(¯ x) ∶= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ φ(¯ x) if φ ∈ FO ∃¯ z (¯ x ⊆ ¯ z ∧ φ(¯ z))

• therwise

Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 5 / 10

Te fragment. Take  – stratified

S∃

 = S∀  = S ∶∶= FO

S∃

i+ ∶∶=

⊆ atoms, Si φ ∨ η ∃¯ xφ for φ, η ∈ Si+ φ♭ ∧ η ∀¯ xφ♭ for φ ∈ Si, η ∈ Si+ S∀

i+ ∶∶=

⊆ atoms, Si φ♭ ∨ η ∃¯ xφ♭ for φ ∈ Si, η ∈ Si+ φ ∧ η ∀¯ xφ for φ, η ∈ Si+ Si+ ∶∶= S∃

i+ ∪ S∀ i+

Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 6 / 10

Te fragment. Take  — with negation

N ∶∶= literals, ⊆ atoms φ ∨ η ∃¯ xφ for φ, η ∈ N φ♭ ∧ η ∀¯ xφ♭ ¬φ♭ with φ♭(¯ x) ∶= ∃¯ z (¯ x ⊆ ¯ z ∧ φ(¯ z)) Semantics A ⊧X φ♭ ∶ ⇐ ⇒ A / ⊧Y φ♭ for all Y ∩ X ≠ ∅ Nk: fragment with fewer than k negations.

Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 7 / 10

N vs Solitaire-GFP

Proposition Winning solitaire games of level k can be described in Nk. On arbitrary structures: Nk ≡ Solitaire-GFP with alternation level k.

Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 8 / 10

Expressiveness

Over arbitrary structures: S collapses to S∃

 = N

F ≡ S ≡ N Ni ⊊ Ni+ for all levels i Over finite structures: N collapses to N F ≡ S ≡ N

Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 9 / 10

Complexity

Proposition

1 Model checking F, S, or N is NL-complete (data complexity). 2 Every NL-property of ordered structures is expressible in F. Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 10 / 10